Question: Determine how many solutions exist for the system of equations. ${10x+2y = -6}$ ${y = -3-5x}$
Solution: Convert both equations to slope-intercept form: ${10x+2y = -6}$ $10x{-10x} + 2y = -6{-10x}$ $2y = -6-10x$ $y = -3-5x$ ${y = -5x-3}$ ${y = -3-5x}$ ${y = -5x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x-3}$ ${y = -5x-3}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${10x+2y = -6}$ is also a solution of ${y = -3-5x}$, there are infinitely many solutions.